11.9. Adadelta¶ Open the notebook in SageMaker Studio Lab
Adadelta是AdaGrad的另一种变体（ 11.7节）， 主要区别在于前者减少了学习率适应坐标的数量。 此外，广义上Adadelta被称为没有学习率，因为它使用变化量本身作为未来变化的校准。 Adadelta算法是在 (Zeiler, 2012)中提出的。
11.9.1. Adadelta算法¶
简而言之，Adadelta使用两个状态变量，\(\mathbf{s}_t\)用于存储梯度二阶导数的泄露平均值，\(\Delta\mathbf{x}_t\)用于存储模型本身中参数变化二阶导数的泄露平均值。请注意，为了与其他出版物和实现的兼容性，我们使用作者的原始符号和命名（没有其它真正理由让大家使用不同的希腊变量来表示在动量法、AdaGrad、RMSProp和Adadelta中用于相同用途的参数）。
以下是Adadelta的技术细节。鉴于参数du jour是\(\rho\)，我们获得了与 11.8节类似的以下泄漏更新：
(11.9.1)¶\[\begin{aligned}
\mathbf{s}_t & = \rho \mathbf{s}_{t-1} + (1 - \rho) \mathbf{g}_t^2.
\end{aligned}\]
与 11.8节的区别在于，我们使用重新缩放的梯度\(\mathbf{g}_t'\)执行更新，即
(11.9.2)¶\[\begin{split}\begin{aligned}
\mathbf{x}_t & = \mathbf{x}_{t-1} - \mathbf{g}_t'. \\
\end{aligned}\end{split}\]
那么，调整后的梯度\(\mathbf{g}_t'\)是什么？我们可以按如下方式计算它：
(11.9.3)¶\[\begin{split}\begin{aligned}
\mathbf{g}_t' & = \frac{\sqrt{\Delta\mathbf{x}_{t-1} + \epsilon}}{\sqrt{{\mathbf{s}_t + \epsilon}}} \odot \mathbf{g}_t, \\
\end{aligned}\end{split}\]
其中\(\Delta \mathbf{x}_{t-1}\)是重新缩放梯度的平方\(\mathbf{g}_t'\)的泄漏平均值。我们将\(\Delta \mathbf{x}_{0}\)初始化为\(0\)，然后在每个步骤中使用\(\mathbf{g}_t'\)更新它，即
(11.9.4)¶\[\begin{aligned}
\Delta \mathbf{x}_t & = \rho \Delta\mathbf{x}_{t-1} + (1 - \rho) {\mathbf{g}_t'}^2,
\end{aligned}\]
和\(\epsilon\)（例如\(10^{-5}\)这样的小值）是为了保持数字稳定性而加入的。
11.9.2. 代码实现¶
Adadelta需要为每个变量维护两个状态变量，即\(\mathbf{s}_t\)和\(\Delta\mathbf{x}_t\)。这将产生以下实现。
%matplotlib inline
from mxnet import np, npx
from d2l import mxnet as d2l
npx.set_np()
def init_adadelta_states(feature_dim):
s_w, s_b = np.zeros((feature_dim, 1)), np.zeros(1)
delta_w, delta_b = np.zeros((feature_dim, 1)), np.zeros(1)
return ((s_w, delta_w), (s_b, delta_b))
def adadelta(params, states, hyperparams):
rho, eps = hyperparams['rho'], 1e-5
for p, (s, delta) in zip(params, states):
# In-placeupdatesvia[:]
s[:] = rho * s + (1 - rho) * np.square(p.grad)
g = (np.sqrt(delta + eps) / np.sqrt(s + eps)) * p.grad
p[:] -= g
delta[:] = rho * delta + (1 - rho) * g * g
%matplotlib inline
import torch
from d2l import torch as d2l
def init_adadelta_states(feature_dim):
s_w, s_b = torch.zeros((feature_dim, 1)), torch.zeros(1)
delta_w, delta_b = torch.zeros((feature_dim, 1)), torch.zeros(1)
return ((s_w, delta_w), (s_b, delta_b))
def adadelta(params, states, hyperparams):
rho, eps = hyperparams['rho'], 1e-5
for p, (s, delta) in zip(params, states):
with torch.no_grad():
# In-placeupdatesvia[:]
s[:] = rho * s + (1 - rho) * torch.square(p.grad)
g = (torch.sqrt(delta + eps) / torch.sqrt(s + eps)) * p.grad
p[:] -= g
delta[:] = rho * delta + (1 - rho) * g * g
p.grad.data.zero_()
%matplotlib inline
import tensorflow as tf
from d2l import tensorflow as d2l
def init_adadelta_states(feature_dim):
s_w = tf.Variable(tf.zeros((feature_dim, 1)))
s_b = tf.Variable(tf.zeros(1))
delta_w = tf.Variable(tf.zeros((feature_dim, 1)))
delta_b = tf.Variable(tf.zeros(1))
return ((s_w, delta_w), (s_b, delta_b))
def adadelta(params, grads, states, hyperparams):
rho, eps = hyperparams['rho'], 1e-5
for p, (s, delta), grad in zip(params, states, grads):
s[:].assign(rho * s + (1 - rho) * tf.math.square(grad))
g = (tf.math.sqrt(delta + eps) / tf.math.sqrt(s + eps)) * grad
p[:].assign(p - g)
delta[:].assign(rho * delta + (1 - rho) * g * g)
%matplotlib inline
import warnings
from d2l import paddle as d2l
warnings.filterwarnings("ignore")
import paddle
def init_adadelta_states(feature_dim):
s_w, s_b = paddle.zeros(shape=(feature_dim, 1)), paddle.zeros(shape=(1, ))
delta_w, delta_b = paddle.zeros(shape=(feature_dim, 1)), paddle.zeros(shape=(1, ))
return ((s_w, delta_w), (s_b, delta_b))
def adadelta(params, states, hyperparams):
a = []
rho, eps = hyperparams['rho'], 1e-5
for p, (s, delta) in zip(params, states):
with paddle.no_grad():
# In-placeupdatesvia[:]
s[:] = rho * s + (1 - rho) * paddle.square(p.grad)
g = (paddle.sqrt(delta + eps) / paddle.sqrt(s + eps)) * p.grad
p[:] -= g
delta[:] = rho * delta + (1 - rho) * g * g
p.grad.zero_()
a.append(p)
return a
对于每次参数更新，选择\(\rho = 0.9\)相当于10个半衰期。由此我们得到：
data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(adadelta, init_adadelta_states(feature_dim),
{'rho': 0.9}, data_iter, feature_dim);
loss: 0.243, 0.101 sec/epoch
data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(adadelta, init_adadelta_states(feature_dim),
{'rho': 0.9}, data_iter, feature_dim);
loss: 0.243, 0.014 sec/epoch
data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(adadelta, init_adadelta_states(feature_dim),
{'rho': 0.9}, data_iter, feature_dim);
loss: 0.243, 0.148 sec/epoch
data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(adadelta, init_adadelta_states(feature_dim),
{'rho': 0.9}, data_iter, feature_dim);
loss: 0.242, 0.059 sec/epoch
为了简洁实现，我们只需使用高级API中的Adadelta算法。
d2l.train_concise_ch11('adadelta', {'rho': 0.9}, data_iter)
loss: 0.243, 0.103 sec/epoch
trainer = torch.optim.Adadelta
d2l.train_concise_ch11(trainer, {'rho': 0.9}, data_iter)
loss: 0.243, 0.013 sec/epoch
# adadeltaisnotconvergingatdefaultlearningrate
# butit'sconvergingatlr=5.0
trainer = tf.keras.optimizers.Adadelta
d2l.train_concise_ch11(trainer, {'learning_rate':5.0, 'rho': 0.9}, data_iter)
loss: 0.244, 0.101 sec/epoch
trainer = paddle.optimizer.Adadelta
d2l.train_concise_ch11(trainer, {'rho': 0.9}, data_iter)
loss: 0.268, 0.031 sec/epoch
11.9.3. 小结¶
Adadelta没有学习率参数。相反，它使用参数本身的变化率来调整学习率。
Adadelta需要两个状态变量来存储梯度的二阶导数和参数的变化。
Adadelta使用泄漏的平均值来保持对适当统计数据的运行估计。
11.9.4. 练习¶
调整\(\rho\)的值，会发生什么？
展示如何在不使用\(\mathbf{g}_t'\)的情况下实现算法。为什么这是个好主意？
Adadelta真的是学习率为0吗？能找到Adadelta无法解决的优化问题吗？
将Adadelta的收敛行为与AdaGrad和RMSProp进行比较。